Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries

Finite-element method for time-dependent Maxwell's equations based on an explicit-magnetic-field scheme

Hybrid finite element (FE) solutions of transient electromagnetic field are proposed. Time dependent Maxwell's curl equations are solved separately using edge elements for the electric field approximation and nodal-based elements for the magnetic field approximation. Two different formulations are presented: the first adopts a unique mesh, the second two complementary FE meshes. The solution of the resulting equations systems is performed by explicit methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In this paper, we propose a new time-domain finite element method for solving the time dependent Maxwell's equations coupled with the Lorentz metamaterial model. The Lorentz metamaterial Maxwell's equations are much more complicated than the standard Maxwell's equations in free space. Our fully discrete scheme uses edge elements to approximate the unknowns in space, and uses the leap-frog scheme in time discretization. Numerical stability and the optimal error estimate in the $$L^2$$L2 norm are proved for our proposed scheme. Extensive numerical results are presented to confirm the theoretical analysis and applications of our scheme to model many interesting phenomena happened when wave propagates in the Lorentz metamaterials. Examples include the convergence effect happened in the concave lenses formed by the negative refraction index metamatrials, and total reflection and total transmission observed in the zero index metamaterials.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

In this paper, we consider the time-dependent Maxwell's equations when Cole–Cole dispersive medium is involved. The Cole–Cole model contains a fractional time derivative term, which couples with the standard Maxwell's equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1–26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572–591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533–1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12–20], we propose two fully discrete mixed finite element schemes for the Cole–Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.

A Fourier penalty method for solving the time-dependent Maxwell's equations in domains with curved boundaries

Biorthogonal interpolating functions have been applied to electromagnetic field modeling through the wavelet-collocation technique in the time domain, yielding a versatile first-principle algorithm for the solution of time dependent Maxwell's equations with inhomogeneous media. The resulting technique is rigorous and maintains high accuracy, while, by virtue of its sub-gridding capability, significant reduction of the computational requirement has been obtained particularly for the modeling of electrically large dielectric waveguides. The proposed technique has been applied to a two dimensional dielectric waveguide beam splitter, with dimensions typical of optical waveguide systems, and improvement in the computational efficiency has been demonstrated.

Maxwell equations are easily resolved when the computational domain is convex or with a smooth boundary, but if on the contrary it includes geometrical singularities, the electromagnetic field is locally unbounded and globally hard to compute. The challenge is to find out numerical methods which can capture the EM field accurately. Numerically speaking, it is advised, while solving the coupled Maxwell-Vlasov system, to compute a continuous approximation of the field. However, if the domain contains geometrical singularities, continuous finite elements span a strict subset of all possible fields, which is made of the H-regular fields. In order to recover the total field, one can use additional ansatz functions or introduce a weight. The first method, known as the singular complement method [4, 3, 14, 2, 9, 15, 16] works well in 2D and 2D 1 2 geometries and the second method, known as the weighted regularization method [13] works in 2D and 3D. In this contribution, we examine some recent developments of the latter method to solve instationary Maxwell equations and we provide numerical results.

Quaternion analysis of time-dependent Maxwell's equations in the presence of electric and magnetic charges has been developed and the solutions for the classical problem of moving charges (electric and magnetic) are obtained in a unique, simple and consistent manner.

Time-dependent current source identification for numerical simulations of Maxwell's equations

A Reflectionless Sponge Layer Absorbing Boundary Condition for the Solution of Maxwell's Equations with High-Order Staggered Finite Difference Schemes

Self-consistent two-dimensional modeling of a steady microwave discharge initiated at the end of the central electrode in nitrogen is presented. The discharge parameters are calculated at a gas pressure of 1 Torr and incident power of 30 W. The computational model includes the time-dependent Maxwell's equations, the balance equations describing the kinetics of charged and neutral plasma particles and the time-independent homogeneous Boltzmann equation for electrons. The processes involving vibrationally excited ground state molecular nitrogen are taken into account by the well-known analytic expression for the vibrational distribution of molecules in the diffusion approximation. It is shown that the spatially non-uniform microwave field causes the difference in plasma particles kinetics in different parts of the discharge. Results of numerical simulations for space distribution of electronically excited molecules have been found in good qualitative agreement with those taken from spectral measurements of first and second positive systems of nitrogen. Results confirm the concept according to which such a discharge comprises a self-sustained and a non-self-sustained discharge.

The time‐dependent Maxwell equations are one of the most important approaches to describing dynamic or wide‐band frequency electromagnetic phenomena. A sequential finite‐volume, characteristic‐based procedure for solving the time‐dependent, three‐dimensional Maxwell equations has been successfully implemented in Fortran before. Due to its need for a large memory space and high demand on CPU time, it is impossible to test the code for a large array. Hence, it is essential to implement the code on a parallel computing system. In this paper, we discuss an efficient and scalable parallelization of the sequential Fortran time‐dependent Maxwell equations solver using High Performance Fortran (HPF). The background to the project, the theory behind the efficiency being achieved, the parallelization methodologies employed and the experimental results obtained on the Cray T3E massively parallel computing system will be described in detail. Experimental runs show that the execution time is reduced drastically through parallel computing. The code is scalable up to 98 processors on the Cray T3E and has a performance similar to that of an MPI implementation. Based on the experimentation carried out in this research, we believe that a high‐level parallel programming language such as HPF is a fast, viable and economical approach to parallelizing many existing sequential codes which exhibit a lot of parallelism. Copyright © 2003 John Wiley &amp; Sons, Ltd.

The precise description electron-hole distribution and their quasi-temperatures are important to understand the initial stage of the laser processing. The multi-photon absorption and tunnel excitation processes are the crucial electron excitation processes in the semiconductors under the femtosecond laser pulse. Since these processes are non-linear and/or non-perturbative, we have to treat the dynamics of the electron and electromagnetic fields simultaneously. Recently, we have developed the first-principle numerical program SALMON, which combining time-dependent density functional theory (TDDFT) and the Maxwell's equation [1]. We can simulate the electron excitation process around the surface in the first-principle calculation.

In dimension three, we show the existence of weak solutions (u, H, E) to the Landau-Lifshitz equation coupled with the time-dependent Maxwell equation such that u is Holder continuous away from a closed set Σ that has locally finite 3-dimensional parabolic Hausdorff measure. For two reduced Maxwell equations, Holder continuity of ∇u away from Σ is also established.